You can't harm your chip by overloading it with too heavy calculations or something like that.

The only way in which you can damage it is by overloading the pins with too much current.

This could happen happen if you short an output or apply an external voltage where it doesn't belong. Or of course by connecting GND and VCC the wrong way around or applying too much voltage to those pins.

And one thing that gets often forgotten: ESD. In my experience the Atmel (or should I say Microchip) parts are very resistant against ESD, but others might have other experiences there.

You mention only DMMs and an Oscilloscope. As long as you don't make short circuits, you should be safe as these devices usually don't source any current and sink very little.

And concerning the fuses, just be careful. Most things can be fixed with a little trickery though.

Just a little recommendation should you ever decide to buy more hardware: For beginners, the cheap Arduino Uno knockoffs with the AtMega in the socket are very handy. The boards are dirt cheap so if you damage it, it doesn't hurt that much. And if it's "only" the AtMega that's burnt, you can replace just the chip for even cheaper.

-Patrick

Edit: One more thing: You asked for the safest way to experiment: I don't really recommend it, but you can just leave all ports as inputs and toggle only the PORT register. This switches the pull-up on and off, that should be enough to monitor the pin as long as you also provide a weak(!, >100k) pull-down resistor. The advantage is that a short circuit doesn't do any harm at all, but the problem is that the pins have a small parasitic capacitance, which leads to an RC circuit. The time constant might just be too long to see anything useful on the scope, and you can't even reliably drive the input of another chip.

"Some people die at 25 and aren't buried until 75." -Benjamin Franklin

...avrdude...Is it possible to harm the uC by accident using software only...

Just be very very sure that you understand the role of the various 'fuses' that can be set within the processor. Especially those related to clock selection. Whilst you can't kill the processor, you can certainly make it go deaf and getting it back is a right PITA.

"This forum helps those that help themselves."

"If you think you need floating point to solve the problem then you don't understand the problem. If you really do need floating point then you have a problem you do not understand."

All the UNO R3 I have use the 16U2, not the 32U4. Are you sure yours is a 32U4? For that matter, are you sure you've got an UNO R3? The Leonardo uses the 32U4, and >>only<< that. It's a single-chip board.

If you've got at 16U2, then the firmware you want can be found in the firmware folder, somewhere in your Arduino software folder. The file you want is:

Arduino-COMBINED-dfu-usbserial-atmega16u2-Uno-Rev3.hex

"Experience is what enables you to recognise a mistake the second time you make it."

System wide configuration file is "/usr/local/arduino-1.8.3/hardware/tools/avr/etc/avrdude.conf"
User configuration file is "/home/mccramer/.avrduderc"
User configuration file does not exist or is not a regular file, skipping

A chip erase restores all of flash (and EEPROM, provided EESAVE fuse is not programmed) to 0xFF. That is, all the bits are restored to 1s.

Programming consists of changing those 1s to 0s. Programming cannot change a 0 back to a 1. Only an erase operation can do that.

So, if you program without first erasing, flash will contain the bitwise-and of what was there before, and what you programmed over top.

Take an existing byte value of 0xAA. That's binary 0b10101010. Now program it with 0x55, which is binary 0b01010101

current: 10101010

new: 01010101

-----------------

and: 00000000

So programming without first erasing will result in flash containing neither the old contents, nor the desired new contents, but something entirely different... >>unless<< the new content is identical to the old content.

"Experience is what enables you to recognise a mistake the second time you make it."